Variants of Classical Set Theory and their Applications

Professor Peter Aczel

Classical Axiomatic Set Theory, as formalised in ZFC; i.e. Zermelo
Fraenkel set theory with the Axiom of Choice, has been used, through
much of this century, as the foundational theory for modern pure
mathematics. This central role for ZFC is based on the fact that all
the mathematical objects needed can be coded in purely set theoretical
terms and their properties can be proved from the ten or so axioms of
ZFC.

For various reasons many other systems of set theory have been studied
by logicians and others. In Manchester three kinds of variants have received
particular attention. These are

Hyperset Theory

Generalised Set Theory

Constructive Set Theory

Hyperset Theory

This is axiomatic set theory, modified by dropping the Axiom of
Foundation and instead adding some of a variety of possible axioms
that assert the existence of non-well-founded sets. While the Axiom of
Foundation is usually thought to be true for the standard
iterative-combinatorial conception of set in which sets are thought of
as being `formed' out of their elements, the axiom plays very little
role in the coding of mathematical objects. In the last decade or two
non-well-founded sets have turned out to be useful in allowing simple
ways to code various kinds of circular and non-well-founded
objects. This has given rise to some elegant mathematics and the
application of non-well-founded sets to linguistics, computer science
and philosophy, more specifically, in linguistics to situation theory
(a mathematical theory that gives a foundation for the situation
semantics approach to the semantics of natural language), in computer
science to process algebra and final semantics, and in philosophy to an
approach to the Liar Paradox.

Generalised Set Theory

This is axiomatic set theory, modified to allow for other kinds of
objects besides pure sets. The simplest modification is reasonably
familiar. This is to allow for atoms (also called urelemente). These
are objects that have no internal structure. As well as sets having
sets as elements they may have atoms also as elements. But it is also
possible to allow for non-sets that do have internal structure. For
example axiomatic set theory can be modified so as to allow for a
primitive notion of ordered pair. So given any two objects a,b of the
universe, sets, atoms or whatever, as well as being allowed to form
the unordered pair set {a,b}, it can be allowed to form a new object
(a,b), called the ordered pair of a,b. This object is not a set but,
like {a,b} is a structured object having internal components a,b.
More elaborate kinds of structured objects that are not sets can also
be allowed.

Constructive Set Theory

Instead of changing the non-logical axioms of axiomatic set theory so
as to allow for non-well-founded sets or for non-sets of various kinds
we may consider changing the logic. One possibility is to replace
classical logic by intuitionistic logic. Provided that the
non-logical axioms of axiomatic set theory are carefully formulated
the resulting set theory has been called Intuitionistic Set Theory.
Constructive Set Theory is intended to be a set theoretical approach
to constructive mathematics. Intuitionistic Set Theory would seem to
be too strong to be taken to be an axiomatic constructive set theory.
Various much weaker subsystems seem to be more appropriate. In
particular there has been a good deal of attention focused on an
axiom system CZF.